help với ,bài này cân=điểm rơi thế nào ạ
Áp dụng BĐT Cauchy-Schwarz và AM-GM ta có:
\(P=\dfrac{2a}{\sqrt{1+a^2}}+\dfrac{b}{\sqrt{1+b^2}}+\dfrac{c}{\sqrt{1+c^2}}\)
\(=\dfrac{2a}{\sqrt{ab+bc+ca+a^2}}+\dfrac{b}{\sqrt{ab+bc+ca+b^2}}+\dfrac{c}{\sqrt{ab+bc+ca+c^2}}\)
\(=\dfrac{2a}{\sqrt{\left(a+b\right)\left(a+c\right)}}+\dfrac{b}{\sqrt{\left(b+c\right)\left(a+b\right)}}+\dfrac{c}{\sqrt{\left(a+c\right)\left(b+c\right)}}\)
\(=\sqrt{\dfrac{2a}{a+b}\cdot\dfrac{2a}{a+c}}+\sqrt{\dfrac{2b}{a+b}\cdot\dfrac{b}{2\left(b+c\right)}}+\sqrt{\dfrac{2c}{a+c}\cdot\dfrac{c}{2\left(b+c\right)}}\)
\(\le\dfrac{\dfrac{2a}{a+b}+\dfrac{2a}{a+c}+\dfrac{2b}{a+b}+\dfrac{b}{2\left(b+c\right)}+\dfrac{2c}{c+a}+\dfrac{c}{2\left(b+c\right)}}{2}=\dfrac{9}{4}\)
Thế thôi :">
